Chapter 4.35, Problem 3E

Let $X$ be metrizable. Show that the following are equivalent:

(i) $X$ is bounded under every metric that gives the topology of $X$.

(ii) Every continuous function $\varphi =X\to ℝ$ is bounded.

(iii) $X$ is limit point compact.

[Hint: If $\varphi =X\to ℝ$ is a continuous function, then $F\left(x\right)=x\times \varphi \left(x\right)$ is an imbedding of $X$ in $X\times ℝ$. If $A$ is an infinite subset of $X$ having no limit point, let $\varphi$ be a surjection of $A$ onto ${\mathbb{Z}}_{+}$.]